Function generator

ABSTRACT

Approximation of a function relating an independent to a dependent variable. A first resistance furnishes signals corresponding to the independent variable values at a plurality of first taps, a second resistance furnishes dependent variable signals at a corresponding plurality of second taps. A first and second contact arm respectively scan the first and second resistors simultaneously. A feedback circuit has an input terminal connected to the input furnishing the value of independent variable for which the value of dependent variable is to be found. It has a feedback input connected to receive the signals scanned by the first contact arm. Movement of contact arms is stopped when signal at first contact arm is equal to input signal. Signal at second contact arm then equal to desired value of dependent variable.

Schirmer et al.-

14s] Jan.2, 1973 54] FUNCTION GENERATOR [75] Inventors: Gunter'Schirmer, Leinfelden; Edwin Fauser, Sersheim, both of Germany [73]Assignee: Robert Bosch GmbH, Stuttgart, Germany [22] Filed: Sept. 23,1971 [21] Appl. No.: 182,982

[30] Foreign Application Priority Data 3,358,130 12/1967 Mluraetal...235/l97 Primary Examiner Joseph F. Ruggiero Attorney-Michael S.Striker 57 ABSTRACT Approximation of a function relating an independentto a dependent variable. A first resistance furnishes signalscorresponding to theindependent variable values at a plurality of firsttaps, a second resistance furnishes dependent variable signals at acorrespond-v ing plurality of second taps. A first and second contactarm respectively scan the first and second resistors simultaneously. Afeedback circuit has an input terminal connected to the input furnishingthe value of independent variable for which the value of dependentvariable is to be found. It has a feedback input connected to receivethe signals scanned by the first contact arm- Movement of contactarms isstopped when signal at first contact arm is equal to input signal.Signal at second contact arm then equal to desired value of dependentvariable.

22 Claims, 6 Drawing Figures Sept. 24, 1970 Germany ..P 20 47 087.5

[52] U.S. Cl. ..235/l97, 307/229, 3281142 [51] Int. Cl. ..G06q 7/26 [58]Field of Search .235/193, 197, 150.53; 307/229,

[56] References Cited UNITED STATES PATENTS 3,100,839 8/1963 Nathan etal. ..235/197 3,102,951 9/1963 Nathan..... 3,345,505 10/1967 Schmid.....

sum 1 m5 fM/r A TI'ORNE y PATENTEDJMI 2191a SHEET 3 [)F 5 Fig.4

PATENTEDJMI 2 I975 SHEET 5 [IF 5 I I I l I I FUNCTION GENERATORBACKGROUND OF THE INVENTION This invention relates to a functiongenerator for simulating a determined function relating an independentand a dependent variable. In particular, it relates to a circuitfurnishing for each input value of independent variable, the value ofdependent variable related thereto by said determined function. Inparticular, the present invention relates to function generators whereinsuch approximation of the determined function is carried outstep-by-step in accordance with either a linear or other approximation.

Known function generators of this type comprise diode functiongenerators. These have the disadvantage that the determined function isapproximated by the slope of linear portions between two points x,/y,.If one of the points x,/y, is to be changed, then the slope of thestraight line connecting the subsequent and following points, must againbe determined and re-set.

In many cases, this procedure is too complicated and.

time-consuming. Further, since the function-generators are oftenoperated by unskilled labor, an exact approximation of the desiredfunction, or of the particular portion of the curve, does not alwaysresult.

SUMMARY OF THE INVENTION It is an object of the present invention tofurnish a function generator wherein the desired function may be set in,for example, with the aid of calibrated knobs, and wherein theremainderof the operation is then antomatic.

The present invention comprises a function generator for approximating adetermined function relating an independent variable to a dependentvariable. It comprises first and second signal furnishing means,respectively furnishing independent variable signals corresponding tovalues of said independent variable and dependent variable signals, eachrelated to the corresponding one of said independent variable signals bysaid determined function. The first and second scanning means aresimultaneously controlled by closed loop control means. The closed loopcontrol means have a first input connected to receive an input signalsignifying the selected value of independent variable, a feedback inputconnected to said first scanning means, and an output jointlycontrolling said first and second scanning means until said scannedindependent variable signal has a predetermined relationship to saidinput signal. The scanned second signals scanned at this point,correspond to the desired value of dependent variable.

The first and second signal furnishing means comprise, for example,first and second resistance means having voltages corresponding tovalues of X, and Y,, respectively, applied at selected points by use,for example, of calibrated knkbs and furnishing the X, and Y, valuesbetween said selected points. The] scanning means may comprise contactor wiper arms sensing the voltages along said resistance means.

In another embodiment of the present invention, the first and secondscanning means may each comprise a first and second contact armfurnishing sequential values of the independent and dependent variablesignals; The scanning means are controlled by the closed loop controlcircuit in such a manner that the scanning is stopped when the desiredvalue (input signal) of independent variable is within the rangesignified by the first and second scanned independent variable .signal.Three subtraction means are provided in this embodiment, the firstsubtracting the scanned independent variable signals one from the other,the second subtracting the scanned dependent variable signals one fromthe other, and the third subtracting the first scanned independentvariable signal from the input signal. Computing means then compute theslope of the curve as a function of the output signal from the first andsecond subtraction means, and form the product of the slope and theoutput of the third subtraction means. Adder means are then providedwhich add the resulting computer output signal to the first scanneddependent variable signal, thereby furnishing an adder output signalcorresponding to the value of dependent variable associated by saiddetermined function with the input value of independent variable.

The novel features which are considered as characteristic for theinvention are set forth in particular in the appended claims. Theinvention itself, however, both as to its construction and its method ofoperation, together with additional objects and advantages thereof, willbe best understood form the following description of specificembodiments whenread in connection with the accompanying drawing.

BRIEF DESCRIPTION OF THE DRAWING FIG. 1 shows a function generator ofthe present invention using two continuous resistance as signalfurnishing means;

FIG. 2 shows the sistances;

voltage variation across said re- FIG. 3 shows a function generatorwherein said signal furnishing means comprise individual re- DESCRIPTIONOF THE PREFERRED EMBODIMENT A preferred embodiment of the presentinvention will now be described with reference to the drawing. v FIG. 1shows a function generator having a first and second continuousresistance, 11 and 10, respectively. Resistors 11 and 10 are referred toas first and second signalfurnishin'g means, respectively. Theseresistances have connecting terminals, namely resistance 10 has.

connecting terminals 12 through 16 spaced along said resistance, whileresistance 11 has tenninals 17 through 21 again spaced along resistance11. A voltage is applied at each of these terminals. Specifically, avoltage source has a first voltage furnishing terminal and a secondvoltage furnishing terminal, herein indicated as ground. Across thissource of voltage, are connected resistors 22 through 26 and-27 through31. A movable wiper arm connects a selected point on resistances 22through 26 each to one of terminals 12 through 16,

while wiper arms tapping resistors 27 through 31, are

connected to terminals 17 through 21, respectively. This causes theindicated voltage variations shown in FIG. 2 to appear across the tworesistors. The first and second scanning means comprise a contact arm 33movable along resistance 11 and a contact arm 32 movable alongresistance 10, respectively. These contact arms or wiper contacts aremechanically intercoupled and moved simultaneously by output controlmeans, here a motor (preferably a servo motor) 34. The input of theservo motor is connected to the output of a comparator 36 which in turnhas a first input connected to an input terminal 37, and a feedbackinput connected to wiper arm 33. Comparator 36 and motor 34 form theclosed loop control means.

The above-described function generator operates as follows:

The dependent variable signals corresponding to points on the curve Y,to Y, are fixed with the aid of resisters 22 through 26. Resistors 27through 31, or rather the wiper arms associated therewith, serve to fixthe coordinates of points X, to X,,. By fixing these points, the valueof the function between points is automatically fixed electrically overresistors and 11. In order then to obtain the value of dependentvariable corresponding to a particular value of independent variable,the value of independent variable is applied at terminal 37, for examplethrough a potentiometer which is not shown. Input terminal 37 isconnected to the first input of comparator means 36. The closed loopcontrol circuit then serves to drive, via motor 34, the wiper arms 33and 32 until the signal value at wiper arm 33, namely the scanned firstsignal or scanned independent variable signal, corresponds to the inputsignal. At this point, the signal appearing at wiper arm 32 correspondsto the desired value of dependent variable and this value isavailable-at output terminal 35 which is connected to wiper arm 32.

If one point on the curve is to be changed, it is only necessary toreplace either a single resistance or a pair of resistances. Theconnection between the so-replaced resistances is automatically made viathe continuous resistances l0 and 11. Thus adjacent points are notaffected by a change in a particular point. If it is desired to move thewhole curve in either the X or the Y direction, a suitable electricalsignal can simply be added at input terminal 37 or at the outputterminal 35.

Since the servo motor 34 in the function generator according to FIG. 1requires a relatively long time for moving the wiper arms over theresistances until a balance is obtained, this response time may for someapplications be too long. It may be shortened by use of an embodiment asshown in FIG. 3. Here the continuous resistances l0 and l l are replacedby a plurality of series-connected resistance means, namely resistors38. Voltages are again applied at determined points along each other andthe number of such terminals exceeds the number of terminals at whichthe voltages are impressed. For this embodiment, the contact arms areoperated by first step switch means, namely a switch 39, rather than theservo motor 34 of FIG. 1. The step switch means are operated independence upon the output signal of a comparator 36, and the overallfunctioning of the circuit of FIG. 3 is the same as that of the circuitof FIG. 1. Step switch 39 is of course adapted both for forward andreverse motion.

Another embodiment of the present invention is shown in FIG. 4. In thisembodiment, the signal furnishing means are divided into coarseapproximation signal furnishing means are divided into coarseapproximation signal furnishing means and fine approximation signalfurnishing'means. In this embodiment, the means applying predeterminedvoltages which are part of the signal furnishing means for furnishingthe independent variable signals comprise a plurality of resistorsparallel with a voltage source as in FIG. 3, the wiper arm of each ofthese resistors here being denoted by terminals 51, 43,44 and 48. Thecorresponding terminals for the dependent variables are numbered52,70,71 and 49/ Each of the scanning means comprise a pair of contactswhich are positioned by a step switch. control 50 to make contact atsequential independent and dependent variable contacts. In the Figure,the contact ismade to the independent variable signal furnishing meansat terminals 43 and 44 and to the dependent signal furnishing means atcontacts 70 vand 7l. Connected between the two contacts constituting thecoarse scanning means are the fine approximation signal furnishingmeans, namely first and second fine resistance means respectivelyconnected between the scanning contacts of the independent and thedependent variable signal furnishing means. The fine resistance meansmay either be a continuous resistance in which case the scanning meanstherefore are a wiper arm which travels continuously over theresistances, or they may be series-connected resistances, as shown inFIG. 3 (resistances 38) in which case the wiper arm scanning theseries-conv nected resistances is moved by a step switch. The fine theresistance means, namely at terminals 18 through 20 and 13 through 15,respectively. The first or independent variable signals are thenfurnished at terminals 40 which comprise not only terminals 18 through20,

butalso the various connecting points of the individual scanning meansfor the independent variable are denoted by reference numeral 33, whilethe fine scanning means for the dependent variable have a referencenumeral 32. The movable contact arm 32 is connected to output terminal35 while the movable arm 33 is connected to the feedback input of acomparator 36, whose first input is connected to input terminal 37 forreceiving the independent variable signal for which the correspondingdependent variable signal is to be furnished by this arrangement. Theoutput of comparator 36 is again connected to a switch means 39, whichcontrols the movements of arms 32 and 33 in such a manner as to decreasethe difference appearing at the input terminals of comparator 36.Specifically, the switch means move arms 32 and 33 until such time asthe difference at the input of comparator 36 is equal to zero. It isseen that this arrangement has the advantage of requiring muchfewerresistors than the arrangement of FIG. 3, since the fineapproximation signal fumishing means are switched between pairs ofterminals as required, rather than existing between all pairs of thecoarse approximation signal furnishing means terminals, as in FIG. 3.

It is of course possible that the zero difference at the comparatorinputs cannot be obtained for a given setting of the contact pair of thecoarse approximation mines whether the independent variable signal iswithin or outside the range of values signified by the two coarse signalfurnishing contacts at which the coarse scanning means is then situated.In the Figure, the lack of balance would obtain if the independentvariable signal were outside the range of independent variable signalscovered between contacts 43 and 44. In this case, a logic circuit 47would determine that the independent variable signals furnished atterminal 37 is out of the range covered by contacts 43 to 44 and wouldtherefore activate the step switch 50 to move the contact pair 42 toeither terminal 51 and 43, or terminal 44 and 48, depending upon whetherthe independent variable signal at terminal 37 is less than or greaterthan the values covered in the above-mentioned range. In practice, thedirection of the signal furnished by logic circuit 47 to switch means 50maybe determined by which of contacts 68 and 69 are contacted by movablecontact 33. Alternatively, the logic circuit could furnish a signalwhose polarity depends upon the polarity of the output signal ofcomparator 36 and furnish this signal whenever one of the end contacts68 or 69 is touched.

A particular example of the logic circuit and the switching means 50associated therewith is shown in FIG. 6. This Figure will be describednow. It is also useful in conjunction with FIG. 5. Specifically, forFIG. 4 I

only the blocks enclosed in dashed lines and labelled 47 and 50,respectively, are applicable. The block labelled 47 shows a first andsecond comparator, 70, and 71, respectively, each of which has a firstinput terminal connected to terminal 37. The second input tenninal ofcomparator 70 is connected to the end terminal 68 at which the minimumvalue of independent variable for the range is available, while thesecond input terminal of comparator 71 is connected to terminal .69, atwhich the maximum value of independent variable in the particular rangeis available. Comparator 71 yields an output signal when the independentvariable signal, X, is less than the maximum signal available over therange. Comparator 70 yields an output signal when X is less than theminimum value of independent variable available in the range. An ANDgate 72 has a first and second input respectively connected to the firstand second comparator output. This AND gate thus gives an output if theindependent variable is less than the maximum and less than the minimumindependent variable value available in the range. A second AND gate,73, has a first input connected via an inverter to the output ofcomparator 71 and via a further inverter to the output of comparator 70.AND gate 73 therefore yields an output when X is larger than the minimumand larger than the maximum value of independent variable of the range.A first and second monostable multivibrator are respectively connectedto the outputs of AND gate 72, the first AND gate, and AND gate 73, thesecond AND gate means. Thus whenever the independent variable is lessthan the range of independent variable covered by two particularcontacts, the first monostable multivibrator yields a signal, while ifit is greater than the range covered, the second monostablemultivibrator yields a signal. The signals are respectively applied to aswitch means 50. Switch means comprises a first coil connected from theoutput of the first monostable multivibrator to ground, and a secondcoil connected to the output of the second monostable multivibrator toground. A step switch, coarse step switch means, are driven in either afirst or second direction in dependence upon which of the multivibratorsfurnishes the output signal controlling the step switch.

I For example, in response to an output signal from the first monostablemultivibrator, the coarse step switch means would move the pair ofcontacts 42, and thereby the pair of contacts 41, a step to .the left,whereby contacts 42 would be connected from contact 51 to contact 43.Thus this particular arrangement, namely FIG. 4 in conjunction withblocks 47 and 50 of FIG. 6, serves to furnish first a coarseapproximation and secondly a fine approximation signal. The output atterminal 35, after the system has come into balance, is the dependentvariable signal which is related to the independent variable signalapplied at terminal 37 by the function being approximated.

A further embodiment of the present invention is shown in FIG. 5. Thecircuit shown in FIG. 5 operates in such a manner that the initial pointin the curve x,/y, and the slope between this point and a neighboringpoint x,,/y,, is detennined. In order to determine the slope at anyparticular point, it is necessary to subtract the X coordinate of thesecond point from that of the first point, and to subtract the Ycoordinate of the second point from that of the first point.-Terminals57 through and 53 through 56, respectively, again determine thecoordinates, X, Y, of the difierent points of the curve. For example, inthe Figure, terminal 54 contains the value of Y,, terminal 55 the valuefor Y;,, terminal 58 the value X,, and terminal 59 the value X Terminals55 and 54 are connected to the inputs of I second subtraction means 61,while terminals 58 and- 59 are connected via the movable contact arms tothe input terminals of first subtraction means 63. Third subtractionmeans 64 have a first input connected to terminal 37 and a second inputconnected to the contact arm which, as shown in the Figure, makes theconnection to terminal 58. The first, second, and third subtractionmeans have first, second and third subtraction outputs, all connectedto'the inputs of a computer 62. The computer forms the product of theoutput of the second and third subtraction means, and divides thisproduct by the output of the first'subtraction means. This the outputsignal of the computer is k i) txrxn The above output signal of thecomputer is applied to one input of adder means 65 to whose other inputis applied the contact arm making contact with terminal 54, that is theother input corresponds to Y The output function of the adder thus isthe desired value of dependent variable, namely:

This of course is the desired value of dependent variable related to theindependent variable by the function being approximated.

As stated above, the logic circuit denoted by 66 in FIG. is identical tothe logic circuit denoted by 47 in FIG. 4, while the switch means 67 ofFIG. 5 are identical to switch means 50 of FIG. 4. These have beendescribed in detail with reference to FIG. 6. In FIG. 6,

' the connections to subtractors 63 and 64 from the input to the logiccircuit means 66 are also shown.'The description will not be repeatedhere.

The arrangement shown in FIG. 5 has the advantage that no fineapproximation signal furnishing means must be present at all, that isall the resistors interconnected between the various coarseapproximation terminals, may be omitted. Further, theembodiment of FIG.5 has a much shorter response time since the only switching is betweencoarse approximation terminals while the remainder of the operation iscarried out by computer, which of course is almost delay-free.

It is of course to be understood that the various embodiments of thepresent invention shown above, can also be carried out with the use ofsemi-conductor switches, rather than the mechanical switching elementsshown here. This of course again decreases the response time.

Further, the resistances interconnecting the various steps need not belinear resistances, but may vary in either a logarithmic ortrigonometric fashion.

While the invention has been illustrated and described as embodied in aspecific type of signal furnishing means, scanning means, and closedloop control means, it is not intended to be limited to the detailsshown, since various modifications and structural and circuit changesmay be made without departing in any way from the spirit of the presentinvention.

Without further analysis, the foregoing will so fully reveal the gist ofthe present invention that others can by applying current knowledgereadily adapt it for vari ous applications without omitting featuresthat, from the standpoint of prior art, fairly constitute essentialcharacteristics of the generic or specific aspects of this inventionand, therefore, such adaptations should and are intended to becomprehended within the meaning v and range of equivalence of thefollowing claims.

What is claimed as new and desired to be protected by Letters Patent isset forth in the appended claims.

We claim:

1. Function generator for approximating a predetermined functionrelating a dependent variable to an independent variable, comprising, incombination, an input terminal receiving an independent variable signalcorresponding to a selected value of said independent variable; firstsignal furnishing means furnishing a plurality of first signals, each ofthus said first signals corresponding to a determined value of saidindependent variable; second signal furnishing means furnishing acorresponding plurality of second signals, each of said second signalscorresponding to a value of dependent variable related by saidpredetermined function to the value of independent variable associatedwith the corresponding one of said first signals; first scanning meansscanning said first signal furnishing means and furnishing a pluralityof scanned first signals; second scanning means connected to said firstscanning means for simultaneous scanning therewith, said second scanningmeans scanning said second signal furnishing means and furnishing aplurality of scanned second signals, each related to the correspondingone of said scanned first signals by said determined function; andclosed loop control means having a first input connected to saidinputterminal, a feedback input connected to said first scanning means,and output control means responsive to the difference between saidsignals applied at said first and feedback inputs, for stopping saidfirst and second scanning means when said difference is a predetermineddifference, said second scanning means thereby furnishing a scannedsecond signal signifying the value of dependent variable related by saiddetermined function to said selected value of independent variable.

2. An arrangement as set forth in claim 1, wherein said first and secondsignal furnishing means comprise voltage furnishing means; and first andsecond resistance means, each having at least a first and secondconnecting terminal connected to said voltage furnishing means.

3. An arrangement as set forth in claim 2, wherein said first and secondscanning means respectively comprise first and second wiper means,mechanically intercoupled; and wherein said output control meanscomprise motor means moving said mechanically intercoupled first andsecond wiper means along said first and second resistance means.

4. An arrangement as set forth in claim 3, wherein said closed loopcontrol means comprise comparator -means furnishing a comparator outputsignal in first wiper means to said feedback input of said com paratormeans.

6. An arrangement as set forth in claim 1, wherein said first and secondsignalfurnishing means comprise, respectively, a first and secondplurality of series-connected resistance means, each of said pluralityof series-connected resistance means having a plurality of connectingterminals; means applying predetermined voltages at selected ones ofsaid connecting terminals of said first and second plurality ofseries-connected resistance means; wherein said first and secondscanning means comprise, respectively, a first and second movablecontact; wherein said closed loop control means comprise comparatormeans having a first input connected to said input terminal, a feedbackinput connected to said first movable contact, and a comparator output;and first switch means connected to said comparator output for movingsaid first and second movable contacts along said connecting terminalsuntil the difference between the signal at said first input and saidfeedback inputis said predetermined difference.

7. An arrangement as set forth in claim 6, wherein said predetermineddifference is zero.

8. An arrangement as set forth in claim 2, wherein said voltagefurnishing means comprise a source of voltage having a first and secondvoltage furnishing terminal; a plurality of third resistance meansconnected in parallel with said source of voltage, each of said thirdresistance means having a movable arm; and means connecting each of saidmovable arms to a corresponding one of said connecting terminals.

9. An arrangement as set forth in claim 8, wherein each of said thirdresistance means is a low resistance.

10. An arrangement as set forth in claim 1, wherein said first andsecond signal furnishing means respectively comprise first and secondcoarse approximation signal furnishing means furnishing respectively aplurality of first and second coarse approximation signals; furthercomprising first and second fine approximation signal furnishing meanseach having a first and second end terminal and intermediate terminals,and furnishing fine approximation signals at said intermediate terminalsin correspondence to signals applied at said first and second endterminals; wherein said first and second scanning means respectivelycomprise first and second coarse scanning means applying, respectively,successive ones of said first and second coarse approximation signals tosaid end terminals of said first and second fine approximation signalfurnishing means, and first and second fine scanning means respectivelyscanning said first and second fine approximation signal furnishingmeans; and wherein said closed loop control means comprise coarse closedloop control means moving said coarse approximation scanning means untilsaid independent variable signal is within the range of said fineapproximation signals furnished by said first fine'approximation signalfurnishing means, and wherein said closed loop control means furthercomprise fine closed loop control means moving said first and secondfine scanning means until said independent variable signal has apredetermined relationship to said fine approximation signal of saidfirst fine approximation signal furnishing means.

11. An arrangement as set forth in claim 10, wherein said first andsecond coarse approximation signal furnishing means comprise meansfurnishing predetermined voltages at corresponding voltage terminals;wherein said first and second scanning means comprise, respectively, afirst and second pair of mechanically intercoupled contacts, each ofsaid pairs'of contacts being adapted to contact simultaneouslysequential ones of said voltage terminals; wherein said first and secondfine approximation signal furnishing means each comprises correspondingfine resistance means; and wherein said coarse closed loop control meanscomprise logic circuit means responsive to said independent variablesignal and said signals at said first and second end terminal of saidfirst fine approximation signal furnishing means, and furnishing a firstlogic signal signifying an independent variable signal exceeding themaximum one of said fine approximation signals, a second logic signalsignifying an independent variable signal less than the minimum of saidfine approximation signals; and coarse step switch means for moving saidpairs of contacts in afirst or second direction in response to saidfirst and second logic signal respectively.

12. An arrangement as set forth in claim 11, wherein said logic circuitmeans comprise first logic comparator means furnishing a first logiccomparator output signal when said independent variable signal is lessthan the maximum one of said fine approximation signals;

second logic comparator means furnishing a second' logic comparatoroutput signal when said independent variable signal is less than theminimum one of said fine approximation signals; first AND gate meanshaving a first input connected to the output of said first logiccomparator means and a second input connected to the output of saidsecond logic comparator means and a firstAND gate output;-second ANDgate means having a first input connected to the output of said firstlogic comparator means, a second input connected to the output of saidsecond logic comparator means and a second AND gate output; invertermeans connected between said first logic comparator output and saidfirst input of said second AND gate; further inverter means connectedbetween said second logic compara tor output and said second input ofsaid second AND gate means; and first and second monostablemultivibrator means respectively connected to said first and second ANDgate outputs.

13. An arrangement as set forth in claim 11, wherein said fine closedloop control means comprise comparator means having a first inputconnected to said input terminal and a second input connected to saidfirst fine scanning means, said comparator means furnishing a comparatoroutput signal at a comparator output; and servo motor means connected tosaid comparator output and controlling said first and second fineapproximation scanning means as a function of said comparator outputsignal.

14. An arrangement as set forth in claim 11, wherein I said logiccircuit means comprise means responsive to the scanning of said endterminals by said first fine scanning means and furnishing a first logicsignal when said first scanning means scans said first end terminal anda second logic signal when said first scanning means scans said secondend terminal.

15. An arrangement as set forth in claim 11, wherein said logic circuitmeans comprise circuits responsive to the scanning of said first andsecond end terminals by said first scanning means and furnishing saidfirst and second logic signal in dependence upon the sign of saidcomparator output signal. v

16. An arrangement as set forth in claim 2, wherein said secondresistance means comprise linear resistance means.

17. An arrangement as set forth in claim 2, wherein said secondresistance means comprise resistance meansvarying logarithmetically.

18. An arrangement as set forth in claim 2, wherein said secondresistance means comprise trigonemetrically varying resistance means.

19. An arrangement as set forth in claim 1, wherein said first scanningmeans comprise means simultaneously scanning a first and secondconsecutive one of said first signals and furnishing an X, and X scannedsignal in response thereto; wherein said second means when the value ofindependent variable corresponding to said independent variable signalhas an I and third subtraction output signal at, respectively, a

first, second, and third subtraction output; further comprising computermeans connected to said first, second, and third subtraction outputs.

20. An arrangement as set forth in claim 19, wherein said firstsubtraction signal corresponds to X X, wherein said second subtractionsignal corresponds to Y Y and wherein said third subtraction signalcorresponds to X X,, where X signifies said independent variable signal.

21, An arrangement as set forth in claim 20, wherein said computingmeans comprise division means dividing said second subtraction outputsignal by said first subtraction output signal thereby furnishing adivision signal; and wherein said computing means further com prisemultiplier means multiplying said division signal by said thirdsubtraction output signal thereby furnishing a computer output signal.

22. An arrangement as set forth in claim 21, further comprising addermeans having a first input connected to the output of said computermeans and a second input connected to said second scanning means andfumishingan adder output signal corresponding to' Y, added to saidcomputer output signal.

1. Function generator for approximating a predetermined functionrelating a dependent variable to an independenT variable, comprising, incombination, an input terminal receiving an independent variable signalcorresponding to a selected value of said independent variable; firstsignal furnishing means furnishing a plurality of first signals, each ofthus said first signals corresponding to a determined value of saidindependent variable; second signal furnishing means furnishing acorresponding plurality of second signals, each of said second signalscorresponding to a value of dependent variable related by saidpredetermined function to the value of independent variable associatedwith the corresponding one of said first signals; first scanning meansscanning said first signal furnishing means and furnishing a pluralityof scanned first signals; second scanning means connected to said firstscanning means for simultaneous scanning therewith, said second scanningmeans scanning said second signal furnishing means and furnishing aplurality of scanned second signals, each related to the correspondingone of said scanned first signals by said determined function; andclosed loop control means having a first input connected to said inputterminal, a feedback input connected to said first scanning means, andoutput control means responsive to the difference between said signalsapplied at said first and feedback inputs, for stopping said first andsecond scanning means when said difference is a predetermineddifference, said second scanning means thereby furnishing a scannedsecond signal signifying the value of dependent variable related by saiddetermined function to said selected value of independent variable. 2.An arrangement as set forth in claim 1, wherein said first and secondsignal furnishing means comprise voltage furnishing means; and first andsecond resistance means, each having at least a first and secondconnecting terminal connected to said voltage furnishing means.
 3. Anarrangement as set forth in claim 2, wherein said first and secondscanning means respectively comprise first and second wiper means,mechanically intercoupled; and wherein said output control meanscomprise motor means moving said mechanically intercoupled first andsecond wiper means along said first and second resistance means.
 4. Anarrangement as set forth in claim 3, wherein said closed loop controlmeans comprise comparator means furnishing a comparator output signal inresponse to the difference between signals at said first and feedbackinputs; and wherein said motor means is connected to the output of saidcomparator means.
 5. An arrangement as set forth in claim 4, furthercomprising input potentiometer means furnishing said independentvariable signal at an input potentiometer tap; means connecting saidinput potentiometer tap to said first input terminal; and meansconnecting said first wiper means to said feedback input of saidcomparator means.
 6. An arrangement as set forth in claim 1, whereinsaid first and second signal furnishing means comprise, respectively, afirst and second plurality of series-connected resistance means, each ofsaid plurality of series-connected resistance means having a pluralityof connecting terminals; means applying predetermined voltages atselected ones of said connecting terminals of said first and secondplurality of series-connected resistance means; wherein said first andsecond scanning means comprise, respectively, a first and second movablecontact; wherein said closed loop control means comprise comparatormeans having a first input connected to said input terminal, a feedbackinput connected to said first movable contact, and a comparator output;and first switch means connected to said comparator output for movingsaid first and second movable contacts along said connecting terminalsuntil the difference between the signal at said first input and saidfeedback input is said predetermined difference.
 7. An arrangement asset forth in claim 6, wherein said predetermined difference is zero. 8.An arrangement as set fortH in claim 2, wherein said voltage furnishingmeans comprise a source of voltage having a first and second voltagefurnishing terminal; a plurality of third resistance means connected inparallel with said source of voltage, each of said third resistancemeans having a movable arm; and means connecting each of said movablearms to a corresponding one of said connecting terminals.
 9. Anarrangement as set forth in claim 8, wherein each of said thirdresistance means is a low resistance.
 10. An arrangement as set forth inclaim 1, wherein said first and second signal furnishing meansrespectively comprise first and second coarse approximation signalfurnishing means furnishing respectively a plurality of first and secondcoarse approximation signals; further comprising first and second fineapproximation signal furnishing means each having a first and second endterminal and intermediate terminals, and furnishing fine approximationsignals at said intermediate terminals in correspondence to signalsapplied at said first and second end terminals; wherein said first andsecond scanning means respectively comprise first and second coarsescanning means applying, respectively, successive ones of said first andsecond coarse approximation signals to said end terminals of said firstand second fine approximation signal furnishing means, and first andsecond fine scanning means respectively scanning said first and secondfine approximation signal furnishing means; and wherein said closed loopcontrol means comprise coarse closed loop control means moving saidcoarse approximation scanning means until said independent variablesignal is within the range of said fine approximation signals furnishedby said first fine approximation signal furnishing means, and whereinsaid closed loop control means further comprise fine closed loop controlmeans moving said first and second fine scanning means until saidindependent variable signal has a predetermined relationship to saidfine approximation signal of said first fine approximation signalfurnishing means.
 11. An arrangement as set forth in claim 10, whereinsaid first and second coarse approximation signal furnishing meanscomprise means furnishing predetermined voltages at correspondingvoltage terminals; wherein said first and second scanning meanscomprise, respectively, a first and second pair of mechanicallyintercoupled contacts, each of said pairs of contacts being adapted tocontact simultaneously sequential ones of said voltage terminals;wherein said first and second fine approximation signal furnishing meanseach comprises corresponding fine resistance means; and wherein saidcoarse closed loop control means comprise logic circuit means responsiveto said independent variable signal and said signals at said first andsecond end terminal of said first fine approximation signal furnishingmeans, and furnishing a first logic signal signifying an independentvariable signal exceeding the maximum one of said fine approximationsignals, a second logic signal signifying an independent variable signalless than the minimum of said fine approximation signals; and coarsestep switch means for moving said pairs of contacts in a first or seconddirection in response to said first and second logic signalrespectively.
 12. An arrangement as set forth in claim 11, wherein saidlogic circuit means comprise first logic comparator means furnishing afirst logic comparator output signal when said independent variablesignal is less than the maximum one of said fine approximation signals;second logic comparator means furnishing a second logic comparatoroutput signal when said independent variable signal is less than theminimum one of said fine approximation signals; first AND gate meanshaving a first input connected to the output of said first logiccomparator means and a second input connected to the output of saidsecond logic comparator means and a first AND gate output; second ANDgate means having a first input connecteD to the output of said firstlogic comparator means, a second input connected to the output of saidsecond logic comparator means and a second AND gate output; invertermeans connected between said first logic comparator output and saidfirst input of said second AND gate; further inverter means connectedbetween said second logic comparator output and said second input ofsaid second AND gate means; and first and second monostablemultivibrator means respectively connected to said first and second ANDgate outputs.
 13. An arrangement as set forth in claim 11, wherein saidfine closed loop control means comprise comparator means having a firstinput connected to said input terminal and a second input connected tosaid first fine scanning means, said comparator means furnishing acomparator output signal at a comparator output; and servo motor meansconnected to said comparator output and controlling said first andsecond fine approximation scanning means as a function of saidcomparator output signal.
 14. An arrangement as set forth in claim 11,wherein said logic circuit means comprise means responsive to thescanning of said end terminals by said first fine scanning means andfurnishing a first logic signal when said first scanning means scanssaid first end terminal and a second logic signal when said firstscanning means scans said second end terminal.
 15. An arrangement as setforth in claim 11, wherein said logic circuit means comprise circuitsresponsive to the scanning of said first and second end terminals bysaid first scanning means and furnishing said first and second logicsignal in dependence upon the sign of said comparator output signal. 16.An arrangement as set forth in claim 2, wherein said second resistancemeans comprise linear resistance means.
 17. An arrangement as set forthin claim 2, wherein said second resistance means comprise resistancemeans varying logarithmetically.
 18. An arrangement as set forth inclaim 2, wherein said second resistance means comprise trigonemetricallyvarying resistance means.
 19. An arrangement as set forth in claim 1,wherein said first scanning means comprise means simultaneously scanninga first and second consecutive one of said first signals and furnishingan Xi and Xk scanned signal in response thereto; wherein said secondscanning means comprise means simultaneously scanning a first and secondconsecutive one of said second signals and furnishing a Yi and Ykscanned signal in response thereto; wherein said closed loop controlmeans comprise means stopping said scanning means when the value ofindependent variable corresponding to said independent variable signalhas an amplitude less than the amplitude of independent variable signalcorresponding to said Xk scanned signal and greater than the value ofindependent variable corresponding to said Xi scanned signal; furthercomprising first, second and third subtraction means connected,respectively, to said first scanning means, said second scanning means,and said input terminal and said first scanning means, and furnishing afirst, second, and third subtraction output signal at, respectively, afirst, second, and third subtraction output; further comprising computermeans connected to said first, second, and third subtraction outputs.20. An arrangement as set forth in claim 19, wherein said firstsubtraction signal corresponds to Xk - Xi ; wherein said secondsubtraction signal corresponds to Yk - Yi; and wherein said thirdsubtraction signal corresponds to X - Xi, where X signifies saidindependent variable signal.
 21. An arrangement as set forth in claim20, wherein said computing means comprise division means dividing saidsecond subtraction output signal by said first subtraction output signalthereby furnishing a division signal; and wherein said computing meansfurther comprise multiplier means mUltiplying said division signal bysaid third subtraction output signal thereby furnishing a computeroutput signal.
 22. An arrangement as set forth in claim 21, furthercomprising adder means having a first input connected to the output ofsaid computer means and a second input connected to said second scanningmeans and furnishing an adder output signal corresponding to Yi added tosaid computer output signal.